Problem: You have found the following ages (in years) of all $6$ lizards at your local zoo: $ 1,\enspace 2,\enspace 2,\enspace 1,\enspace 3,\enspace 3$ What is the average age of the lizards at your zoo? What is the variance? Round your answers to the nearest tenth. Average age: $ $
Solution: Because we have data for all $6$ lizards at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$. To find the population mean, add up the values of all $6$ ages and divide by $6$. $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\mu} = \dfrac{1 + 2 + 2 + 1 + 3 + 3}{{6}} = {2\text{ years old}} $ Find the squared deviations from the mean for each lizard. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $1$ year $-1$ years $1$ year $^2$ $2$ years $0$ years $0$ years $^2$ $2$ years $0$ years $0$ years $^2$ $1$ year $-1$ years $1$ year $^2$ $3$ years $1$ year $1$ year $^2$ $3$ years $1$ year $1$ year $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean, we can find the population variance $({\sigma^2})$, without introducing any bias, by simply averaging the squared deviations from the mean: $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{1} + {0} + {0} + {1} + {1} + {1}} {{6}} $ $ {\sigma^2} = \dfrac{{4}}{{6}} = {0.67\text{ years}^2} $ The average lizard at the zoo is $2$ years old. The population variance is $0.7$ years $^2$.